RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2000, Volume 34, Issue 1, Pages 51–64 (Mi faa282)

Anisotropic Young Diagrams and Jack Symmetric Functions

S. V. Kerov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We study the Young lattice with the edge multiplicities $\varkappa_\alpha(\lambda,\Lambda)$ arising in the simplest Pieri formula for Jack symmetric polynomials $P_\lambda(x;\alpha)$ with parameter $\alpha$. A new proof of Stanley's $\alpha$-version of the hook formula is given. We also prove the formula
$$\sum_\Lambda (c_\alpha(b)+u)(c_\alpha(b)+v)\varkappa_\alpha(\lambda,\Lambda)\varphi(\Lambda)= (n\alpha+uv)\varphi(\lambda),$$
where $\varphi(\lambda)=\prod_{b\in\lambda}(a(b)\alpha+l(b)+1)^{-1}$ and $c_\alpha(b)$ is the $\alpha$-contents of the new box $b=\Lambda\setminus\lambda$.

DOI: https://doi.org/10.4213/faa282

Full text: PDF file (1286 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2000, 34:1, 41–51

Bibliographic databases:

UDC: 519.217+517.986

Citation: S. V. Kerov, “Anisotropic Young Diagrams and Jack Symmetric Functions”, Funktsional. Anal. i Prilozhen., 34:1 (2000), 51–64; Funct. Anal. Appl., 34:1 (2000), 41–51

Citation in format AMSBIB
\Bibitem{Ker00} \by S.~V.~Kerov \paper Anisotropic Young Diagrams and Jack Symmetric Functions \jour Funktsional. Anal. i Prilozhen. \yr 2000 \vol 34 \issue 1 \pages 51--64 \mathnet{http://mi.mathnet.ru/faa282} \crossref{https://doi.org/10.4213/faa282} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1756734} \zmath{https://zbmath.org/?q=an:0959.05116} \transl \jour Funct. Anal. Appl. \yr 2000 \vol 34 \issue 1 \pages 41--51 \crossref{https://doi.org/10.1007/BF02467066} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000087490500005} 

• http://mi.mathnet.ru/eng/faa282
• https://doi.org/10.4213/faa282
• http://mi.mathnet.ru/eng/faa/v34/i1/p51

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Borodin, A, “Distributions on partitions, point processes, and the hypergeometric kernel”, Communications in Mathematical Physics, 211:2 (2000), 335
2. Regev, A, “S-infinity representations and combinatorial identities”, Transactions of the American Mathematical Society, 353:11 (2001), 4371
3. G. I. Olshanskii, “Probability Measures on Dual Objects to Compact Symmetric Spaces and Hypergeometric Identities”, Funct. Anal. Appl., 37:4 (2003), 281–301
4. S. V. Kerov, “Multidimensional hypergeometric distribution, and characters of the unitary group”, J. Math. Sci. (N. Y.), 129:2 (2005), 3697–3729
5. Olshanski, C, “An introduction to harmonic analysis on the infinite symmetric group”, Asymptotic Combinatorics With Applications To Mathematical Physics, 1815 (2003), 127
6. Olshanski G., “Point processes related to the infinite symmetric group”, Orbit Method in Geometry and Physics - in Honor of A.A. Kirillov, Progress in Mathematics, 213, 2003, 349–393
7. Fulman, J, “Stein's method, Jack measure, and the Metropolis algorithm”, Journal of Combinatorial Theory Series A, 108:2 (2004), 275
8. Kerov, S, “Harmonic analysis on the infinite symmetric group”, Inventiones Mathematicae, 158:3 (2004), 551
9. Lassalle, M, “Jack polynomials and some identities for partitions”, Transactions of the American Mathematical Society, 356:9 (2004), 3455
10. Borodin, A, “Z-measures on partitions and their scaling limits”, European Journal of Combinatorics, 26:6 (2005), 795
11. Fulman, J, “An inductive proof of the berry-esseen theorem for character ratios”, Annals of Combinatorics, 10:3 (2006), 319
12. Fulman, J, “Martingales and character ratios”, Transactions of the American Mathematical Society, 358:10 (2006), 4533
13. Borodin, A, “Giambelli compatible point processes”, Advances in Applied Mathematics, 37:2 (2006), 209
14. Borodin, A, “Markov processes on partitions”, Probability Theory and Related Fields, 135:1 (2006), 84
15. Nekrasov N.A., Okounkov A., “Seiberg-Witten theory and random partitions”, Unity of Mathematics - IN HONOR OF THE NINETIETH BIRTHDAY OF I.M. GELFAND, Progress in Mathematics, 244, 2006, 525–596
16. Marshakov, A, “Extended Seiberg-Witten theory and integrable hierarchy”, Journal of High Energy Physics, 2007, no. 1, 104
17. Strahov, E, “Matrix Kernels for Measures on Partitions”, Journal of Statistical Physics, 133:5 (2008), 899
18. Fulman, J, “Stein's method and random character ratios”, Transactions of the American Mathematical Society, 360:7 (2008), 3687
19. Lassalle, M, “Jack polynomials and free cumulants”, Advances in Mathematics, 222:6 (2009), 2227
20. L. Petrov, “Random walks on strict partitions”, J. Math. Sci. (N. Y.), 168:3 (2010), 437–463
21. Strahov, E, “Z-measures on partitions related to the infinite Gelfand pair (S(2 infinity), H(infinity))”, Journal of Algebra, 323:2 (2010), 349
22. Olshanski G., “Plancherel averages: Remarks on a paper by Stanley”, Electronic Journal of Combinatorics, 17:1 (2010), R43
23. Strahov E., “The z-measures on partitions, Pfaffian point processes, and the matrix hypergeometric kernel”, Advances in Mathematics, 224:1 (2010), 130–168
24. Fulman J., Goldstein L., “Zero Biasing and Jack Measures”, Combinatorics Probability & Computing, 20:5 (2011), 753–762
25. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., “A Form Factor Approach to the Asymptotic Behavior of Correlation Functions in Critical Models”, J. Stat. Mech.-Theory Exp., 2011, P12010
26. Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., “Form Factor Approach to Dynamical Correlation Functions in Critical Models”, J. Stat. Mech.-Theory Exp., 2012, P09001
27. N. A. Slavnov, “Asymptotic expansions for correlation functions of one-dimensional bosons”, Theoret. and Math. Phys., 174:1 (2013), 109–121
28. Bufetov A., “Kerov's Interlacing Sequences and Random Matrices”, J. Math. Phys., 54:11 (2013), 113302
29. Lassalle M., “Class Expansion of Some Symmetric Functions in Jucys-Murphy Elements”, J. Algebra, 394 (2013), 397–443
30. Petrov L., “Sl(2) Operators and Markov Processes on Branching Graphs”, J. Algebr. Comb., 38:3 (2013), 663–720
31. J. Math. Sci. (N. Y.), 215:6 (2016), 755–768
32. Dolega M., Feray V., “Gaussian fluctuations of Young diagrams and structure constants of Jack characters”, Duke Math. J., 165:7 (2016), 1193–1282
33. G. I. Olshanskii, “The Topological Support of the z-Measures on the Thoma Simplex”, Funct. Anal. Appl., 52:4 (2018), 308–310
34. Sergeev A.N., “Super Jack-Laurent Polynomials”, Algebr. Represent. Theory, 21:5 (2018), 1177–1202
35. Sniady P., “Asymptotics of Jack Characters”, J. Comb. Theory Ser. A, 166 (2019), 91–143
36. Dolega M. Sniady P., “Gaussian Fluctuations of Jack-Deformed Random Young Diagrams”, Probab. Theory Relat. Field, 174:1-2 (2019), 133–176
37. Alexander Moll, “Exact Bohr–Sommerfeld Conditions for the Quantum Periodic Benjamin–Ono Equation”, SIGMA, 15 (2019), 098, 27 pp.
•  Number of views: This page: 384 Full text: 129 References: 54